Method and system for utilizing space-time overlays for convolutionally coded systems

ABSTRACT

A communication system for transmitting encoded signals over a communication channel is disclosed. The system includes a transmitter, which has a source that is configured to output a message signal, and an encoder that is configured to generate a code word in response to the message signal. The code word has a construction that is based upon a single dimensional binary code and that specifies a space-time overlay having a predetermined constraint. The transmitter also includes a modulator that is configured to modulate the code word for transmission over the communication channel. Further, the transmitter includes multiple transmit antennas that are configured to transmit the modulated code word over the communication channel. The system also includes a receiver, which may include multiple receive antennas. The receiver is configured to receive the transmitted code word via the multiple receive antennas.

CROSS-REFERENCES TO RELATED APPLICATION

This application is a continuation application of U.S. patentapplication Ser. No. 10/012,950 filed Nov. 7, 2001, entitled “Method andSystem for Utilizing Space-Time Overlays for Convolutionally CodedSystems”. This application is related to, and claims the benefit of theearlier filing date of U.S. Provisional Patent Application (AttorneyDocket PD-200356), Ser. No. 60/249,553, filed Nov. 17, 2000, entitled“Method and Constructions for Space-Time Codes for Block FadingChannels,” the entirety of which is incorporated herein by reference.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The present invention relates to coding in a communication system, andis more particularly related to space-time codes having spatialdiversity and temporal diversity.

2. Discussion of the Background

Given the constant demand for higher system capacity of wirelesssystems, multiple antenna systems have emerged to increase systembandwidth vis-á-vis single antenna systems. In multiple antenna systems,data is parsed into multiple streams, which are simultaneouslytransmitted over a corresponding quantity of transmit antennas. At thereceiving end, multiple receive antennas are used to reconstruct theoriginal data stream. To combat the detrimental effects of thecommunication channel, communication engineers are tasked to developchannel codes that optimize system reliability and throughput in amultiple antenna system.

Almost all digital wireless communication systems employ some form ofchannel coding to protect the raw data from channel noise and multi-pathfading effects. In single transmit antenna systems, channel coding onlyadds temporal redundancy to the raw data aiming to exploit the temporaldiversity provided by time varying wireless fading channels. Theavailability of multiple transmit antennas allows for an additionaldegree of freedom in code design. Space-time coding was introduced in byTarokh et al. [1] as a two dimensional coding paradigm that exploits thespatial diversity provided by multiple transmit antennas in quasi-staticflat fading channels.

To minimize the effects of the communication channel, which typically isRayleigh, space-time codes have been garnered significant attention.Rayleigh fading channels introduce noise and attenuation to such anextent that a receiver may not reliably reproduce the transmitted signalwithout some form of diversity; diversity provides a replica of thetransmitted signal. Space-time codes are two dimensional channel codesthat exploit spatial transmit diversity, whereby the receiver canreliably detect the transmitted signal. Conventional designs ofspace-time codes have focused on maximizing spatial diversity inquasi-static fading channels and fast fading channels. However, realcommunication systems exhibit channel characteristics that are somewherebetween quasi-static and fast fading. Accordingly, such conventionalspace-time codes are not optimized.

Further, other approaches to space-time code design assume that channelstate information (CSI) are available at both the transmitter andreceiver. Thus, a drawback of such approaches is that the designrequires the transmitter and receiver to have knowledge of the CSI,which increases implementation costs because of the need for additionalhardware. Moreover, these approaches view the transmit diversityattending the use of space-time codes as a substitute for timediversity; consequently, such space-time codes are not designed to takeadvantage of other forms of diversity.

Based on the foregoing, there is a clear need for improved approachesfor providing space-time codes that can be utilized in a multi-inputmulti-output (MIMO) block fading channel. There is also a need to designspace-time codes that can exploit spatial diversity as well as timediversity. There is also a need to improve system reliability withoutreducing transmission rate. There is a further need to simplify thereceiver design. Therefore, an approach for constructing space-timecodes that can enhance system reliability and throughput in a multipleantenna system is highly desirable.

SUMMARY OF THE INVENTION

The present invention addresses the above stated needs by providingspace-time overlay codes to optimally exploit the spatial and temporaldiversity available in a communication channel. In an exemplaryembodiment, these space-time overlay codes are implemented to upgradeconvolutionally coded single antenna wireless communication systems. Thealgebraic framework to construct these convolutional space-time overlaysthat achieve full spatial diversity in quasi-static fading channelswithout altering the signal transmitted from the first antenna isdeveloped. For BPSK modulated systems, a general approach forconstructing space-time overlay codes with the same trellis complexityas the code used in the single antenna system is provided. The generalapproach for QPSK modulated systems involves the use of systematic innerspace-time codes that utilize separate soft input/soft output decodersat the receiver. For QPSK modulated systems using rate 1/n binaryconvolutional codes with Gray mapping, an alternative space-timeconstruction with the same trellis complexity as the single dimensionalconvolutional code is developed. The framework for constructingalgebraic space-time overlays, according to an embodiment of the presentinvention, extends to block coded systems.

According to one aspect of the invention, a method for transmittingencoded signals over a communication channel of a communication systemhaving a plurality of transmit antennas and a plurality of receiveantennas is provided. The method includes receiving a message signal,and generating a code word in response to the message signal. The codeword has a construction that is based upon a single dimensional binarycode and that specifies a space-time overlay having a predeterminedconstraint. Under this approach, spatial diversity and temporaldiversity are enhanced, without sacrificing transmission rate.

According to another aspect of the invention, an apparatus for encodingsignals for transmission over a communication channel of a communicationsystem having a plurality of transmit antennas is provided. Theapparatus comprises a source that is configured to output a messagesignal, and an encoder that is configured to generate a code word inresponse to the message signal. The code word has a construction that isbased upon a single dimensional binary code and that specifies aspace-time overlay having a predetermined constraint. The abovearrangement advantageously improves system throughput and systemreliability of a communication system.

According to one aspect of the invention, an apparatus for encodingsignals for transmission over a communication channel of a communicationsystem having a plurality of transmit antennas is provided. Theapparatus includes means for receiving a message signal, and means forgenerating a code word in response to the message signal. The code wordhas a construction that is based upon a single dimensional binary codeand that specifies a space-time overlay having a predeterminedconstraint. The above arrangement advantageously provides increasedsystem capacity.

According to another aspect of the invention, a communication system fortransmitting encoded signals over a communication channel is disclosed.The system includes a transmitter, which has a source that is configuredto output a message signal, and an encoder that is configured togenerate a code word in response to the message signal. The code wordhas a construction that is based upon a single dimensional binary codeand that specifies a space-time overlay having a predeterminedconstraint. The transmitter also includes a modulator that is configuredto modulate the code word for transmission over the communicationchannel. Further, the transmitter includes a plurality of transmitantennas that are configured to transmit the modulated code word overthe communication channel. The system also includes a receiver, whichhas a plurality of receive antennas; the receiver is configured toreceive the transmitted code word via the plurality of receive antennas.The above arrangement advantageously maximizes spatial and temporaldiversity.

According to another aspect of the invention, a waveform signal fortransmission over a communication channel of a communication systemhaving a plurality of transmit antennas and a plurality of receiveantennas is disclosed. The waveform signal includes a code word that hasa construction that is based upon a single dimensional binary code andthat specifies a space-time overlay having a predetermined constraint.The above approach minimizes data transmission errors.

In yet another aspect of the invention, a computer-readable mediumcarrying one or more sequences of one or more instructions fortransmitting encoded signals over a communication channel of acommunication system having a plurality of transmit antennas and aplurality of receive antennas is disclosed. The one or more sequences ofone or more instructions include instructions which, when executed byone or more processors, cause the one or more processors to perform thestep of receiving a message signal. Another step includes generating acode word in response to the message signal. The code word has aconstruction that is based upon a single dimensional binary code andthat specifies a space-time overlay having a predetermined constraint.This approach advantageously provides simplified receiver design.

In yet another aspect of the present invention, an apparatus forreceiving signals over a communication channel of a communication systemhaving a plurality of transmit antennas is provided. The apparatusincludes a demodulator that is configured to demodulate a signalcontaining a code word, wherein the code word has a construction that isbased upon a single dimensional binary code and that specifies aspace-time overlay having a predetermined constraint. The apparatus alsoincludes a decoder that is configured to decode the code word and tooutput a message signal. Under this approach, the effective bandwidth ofthe communication system is increased.

BRIEF DESCRIPTION OF THE DRAWINGS

A more complete appreciation of the invention and many of the attendantadvantages thereof will be readily obtained as the same becomes betterunderstood by reference to the following detailed description whenconsidered in connection with the accompanying drawings, wherein:

FIG. 1 is a diagram of a communication system configured to utilizespace-time codes, according to an embodiment of the present invention;

FIG. 2 is a diagram of an encoder that generates space-time codes, inaccordance with an embodiment of the present invention;

FIG. 3 is a diagram of a decoder that decodes space-time codes,according to an embodiment of the present invention;

FIGS. 4A-4F are graphs of simulation results of the performance ofspace-time codes, according to the embodiments of the present invention;

FIG. 5 is a diagram of a wireless communication system that is capableof employing the space-time codes, according to embodiments of thepresent invention; and

FIG. 6 is a diagram of a computer system that can perform the processesof encoding and decoding of space-time codes, in accordance with anembodiment of the present invention.

DESCRIPTION OF THE PREFERRED EMBODIMENTS

In the following description, for the purpose of explanation, specificdetails are set forth in order to provide a thorough understanding ofthe invention. However, it will be apparent that the invention may bepracticed without these specific details. In some instances, well-knownstructures and devices are depicted in block diagram form in order toavoid unnecessarily obscuring the invention.

Although the present invention is discussed with respect to binaryphase-shift keying (BPSK) and quadrature phase shift keying (QPSK), thepresent invention has applicability to other modulations schemes.

FIG. 1 shows a diagram of a communication system configured to utilizespace-time codes, according to an embodiment of the present invention. Adigital communication system 100 includes a transmitter 101 thatgenerates signal waveforms across a communication channel 103 to areceiver 105. In the discrete communication system 100, transmitter 101has a message source that produces a discrete set of possible messages;each of the possible messages have a corresponding signal waveform.These signal waveforms are attenuated, or otherwise altered, bycommunications channel 103. As a result, receiver 105 must be able tocompensate for the attenuation that is introduced by channel 103. Toassist with this task, transmitter 101 employs coding to introduceredundancies that safeguard against incorrect detection of the receivedsignal waveforms by the receiver 105.

The present invention, according to one embodiment, considers the designof space-time overlays to upgrade single antenna wireless communicationsystems to accommodate multiple transmit antennas efficiently. Anoverlay constraint is defined such that the signal transmitted from thefirst antenna in the upgraded system is the same as that in the singleantenna system, as more fully described below. The signals transmittedfrom the remaining antennas of the transmitter 101 are designedaccording to space-time coding principles to achieve full spatialdiversity in quasi-static flat fading channels 103. For both BPSK andQPSK modulated systems, an algebraic design framework that exploits thestructure of existing single dimensional convolutional codes indesigning overlays that achieve full spatial diversity with minimumadditional decoding complexity at the receiver 105. A concatenatedcoding approach for BPSK overlay design is also developed in which theinner code is an orthogonal block code. This approach yields nearoptimal performance for quasi-static fading channels. Such an approachmay be extended time varying block fading channels.

Conventional space-time code design has not fully considered thelimitations and capabilities of existing single antenna wirelesssystems. On one hand, the optimized physical layer parameters obtainedfrom these traditional designs may not satisfy certain practicalconstraints imposed on the system for example, certain designs requirelarger constellation sizes to achieve the same throughput [2]. On theother hand, such designs did not exploit the single dimensional channelcoding already employed in almost all practical single antenna systems.In contrast, the design of space-time overlays, according to anembodiment of the present invention, provides for upgradingconvolutionally coded single antenna wireless systems to efficientlyaccommodate multiple transmit antennas. The present invention permits analgebraic design approach that utilizes the structure of singledimensional convolutional codes to construct space-time overlays thatachieve full spatial transmit diversity while satisfying a certainoverlay constraint. This constraint ensures that the signal transmittedfrom the first antenna in the upgraded system is the same as that in thesingle antenna system.

For binary phase-shift keying (BPSK) modulated systems with rate k/nbinary convolutional codes, space-time overlays, according to oneembodiment of the present invention, are constructed that preserve thesame trellis complexity of the single dimensional code. For quadraturephase shift keying (QPSK), the general design approach entails the useof systematic inner codes that achieve full diversity, according to oneembodiment of the present invention. This embodiment may imposeadditional complexity required to decode the inner space-time code,relatively to the BSPK system. However, for the special case of QPSKsystems using rate 1/n binary convolutional codes with Gray mapping, aspace-time overlay construction, according to one embodiment of thepresent invention, provides the same trellis complexity as the singledimensional code. Therefore, in most cases, the present inventionpermits use of a space-time maximum likelihood decoder with the sametrellis complexity as a single dimensional decoder.

Although the present invention primarily discusses quasi-static fadingchannels, it is recognized by one of ordinary skill in the art that thepresent invention has applicability to time-varying block fadingchannels as well. Such an extension is based on the framework describedin an article by H. El Gamal and A. R. Hammons Jr., entitled “On theDesign of Algebraic Space-time Codes for Block Fading Channels”[publication data needed], which is incorporated herein by reference inits entirety. One important result in this regard, as detailed below,pertains to the inner orthogonal coding approach and its inability toachieve the maximum possible diversity advantage in such channels.

FIG. 2 shows a diagram of an encoder that generates space-time codes, inaccordance with an embodiment of the present invention. A transmitter200, as mentioned above, possesses a message source 201 that generates ksignals from a discrete alphabet, X′. Encoder 203 then generates signalsfrom alphabet Y to a modulator 205. Modulator 205 maps the encodedmessages from encoder 203 to signal waveforms that are transmitted toL_(t) number of antennas 207, which emit these waveforms over thecommunication channel 103. Accordingly, the encoded messages aresegmented into L_(t) data streams and then simultaneously transmittedover the antennas 207.

FIG. 3 shows a diagram of a decoder that decodes space-time codes,according to an embodiment of the present invention. At the receivingside, a receiver 300 includes a demodulator 301 that performsdemodulation of received signals from transmitter 200. These signals arereceived at multiple antennas 303, which are of a limited number. Thisscenario represents, for example, the down-link of most wireless systemswhereby the number of receive antennas 303 at the terminal is limited bythe weight, size, and battery consumption requirements. Accordingly, thespace-time code design problem in such systems (e.g., 100) presents agreater engineering challenge than that of systems with a large numberof receive antennas 303. In the latter scenario, efficient signalprocessing algorithms can be exploited to separate the signalstransmitted from different antennas at the receiver; this reduces thecode design problem to a single dimensional code design in time-varyingblock fading channels, whereas the environment of the present inventionaccording to one embodiment requires the use of two dimensional codes toaccount for the mutually interfering transmitted signals. Thesealgorithms are described in a paper by H. El Gamal and A. R. HammonsJr., entitled, “The layered space-time architecture: a new perspective”to appear IEEE Trans. Info. Theory, 1999; which is incorporated hereinby reference in its entirety.

After demodulation, the received signals are forwarded to a decoder 305,which attempts to reconstruct the original source messages by generatingmessages, X′. Receiver 300, according to one embodiment of the presentinvention, has a memory 307 that stores channel state information (CSI)associated with the communication channel 103. Conventionalcommunication systems typically require that CSI be available at boththe transmitter and the receiver. By contrast, the present invention,according to one embodiment, does not require CSI at the transmitter200, thus, providing a more robust design.

In a traditional single antenna system, the source generates kinformation symbols from the discrete alphabet X, which are encoded bythe error control code C^((s)) to produce code words of └c₁ ^((s)),c₂^((s)), . . . ,c_(n-1) ^((s)),c_(n) ^((s))┘ length n over the symbolalphabet Y. The modulator mapping function ƒ:Y→Ω then maps the encodedsymbols into constellation points from the discrete complex-valuedsignaling constellation Ω for transmission across the channel. In themulti-antenna system 100, the k information symbols are encoded by thecomposite error control code C to produce code words of length N=nL_(t)over the symbol alphabet Y. The encoded symbols are parsed among L_(t)transmit antennas 207 and, as part of the overlay constraint, mapped bythe same modulator ƒ into constellation points. The modulated streamsfor all antennas 207 are transmitted simultaneously. At the receiver300, there are L_(r) receive antennas 303 to collect the incomingtransmissions. The received baseband signals are subsequently decoded bythe space-time decoder 305. Each spatial channel (the link between onetransmit antenna and one receive antenna) is assumed to experiencestatistically independent flat Rayleigh fading.

A space-time code is defined to include an underlying error control codetogether with a spatial parsing formatter. An L_(t)×n space-time code Cof size M includes an (L_(t)n, M) error control code C and a spatialparser σ that maps each code word vector {overscore (c)}∈ C to anL_(t)×n matrix c whose entries are a rearrangement of those of c. Thespace-time code C is said to be linear if both C and σ are linear. It isassumed that the standard parser maps{overscore (c)}=(c ₁ ⁽¹⁾ ,c ₁ ⁽²⁾ , . . . ,c ₁ ^((L) ^(t) ⁾ ,c ₂ ⁽¹⁾ ,c₂ ⁽²⁾ , . . . ,c ₂ ^((L) ^(t) ⁾ , . . . ,c _(n) ⁽¹⁾ ,c _(n) ⁽²⁾ , . . .,c _(n) ^((L) ^(t) ⁾ ∈Cto the matrix $c = {\begin{bmatrix}c_{1}^{(1)} & c_{2}^{(1)} & \ldots & c_{n}^{(1)} \\c_{1}^{(2)} & c_{2}^{(2)} & \ldots & c_{n}^{(2)} \\\vdots & \vdots & ⋰ & \vdots \\c_{1}^{(L_{t})} & c_{2}^{(L_{t})} & \ldots & c_{n}^{(L_{t})}\end{bmatrix}.}$Base upon the above notation, it is understood that c_(t) ^((i)) is thecode symbol assigned to transmit antenna i at time t. Therefore, theoverlay requirement translates to the following constraint[c ₁ ^((s)) ,c ₂ ^((s)) , . . . ,c _(n-1) ^((s)) ,c _(n) ^((s)) ]=[c ₁⁽¹⁾ ,c ₂ ⁽¹⁾ , . . . ,c _(n-1) ⁽¹⁾ ,c _(n) ⁽¹⁾]  (1)

Assuming s=ƒ(c) is the baseband version of the code word as transmittedacross the channel 103, the following baseband model of the receivedsignal for the overlay system results: $\begin{matrix}{y_{t}^{j} = {{\sqrt{E_{s}}{\sum\limits_{i = 1}^{L_{t}}\quad{\alpha_{t}^{ij}s_{t}^{(i)}}}} + n_{t}^{j}}} & (2)\end{matrix}$where √{square root over (E_(s))} is the energy per transmitted symbol;α_(t) ^(ij) is the complex path gain from transmit antenna i (e.g., 207)to receive antenna j (e.g., 303) at time t, s_(t) ^(i)=ƒ(c_(t) ¹) is thetransmitted constellation point from antenna i at time t; n_(t) ^(j) isthe additive white Gaussian noise sample for receive antenna j at timet. The noise samples are independent samples of zero-mean complexGaussian random variable with variance N₀/2 per dimension. The differentpath gains α_(t) ^(ij) are assumed to be statistically independent. Thefading model of interest is that of a quasi-static flat Rayleigh fadingprocess in which the complex fading gains are constant over the samecode word and are independent from one code word to the next. Channelstate information is assumed to be available a priori only at thereceiver 105.

The diversity advantage of a space-time code is defined as the minimumof the absolute value of the asymptotic slope of the pairwiseprobability of error versus signal-to-noise ratio curve on a log-logscale. The following rank criterion maximizes the spatial diversityadvantage provided by the multiple transmit antenna: for the basebandrank criterion, d=rank(ƒ(c)−ƒ(e)) is maximized over all pairs ofdistinct code words c, e ∈ C. Full spatial transmit diversity isachieved if and only if rank(ƒ(c)−ƒ(e))=L_(t) for all pairs of distinctcode words c, e ∈ C It is noted that in the presence of L_(r) receiveantennas 303, the total diversity advantage achieved by this code inquasi-static fading channels is L_(t)L_(r).

With respect to the design of the space-time overlay, a design frameworkfor full diversity space-time codes that satisfy the overlay constraintaccording to expression (1) and require minimal additional decodingcomplexity over the single dimensional Viterbi decoder used in thesingle antenna system is as follows. For the purpose of explanation, aBPSK modulated system is described. For BPSK modulation, the naturaldiscrete symbol alphabet Y is the field F={0, 1} of integers modulo 2.Modulation is performed by mapping the symbol x ∈ F to the constellationpoint s=ƒ(x)∈{−1, 1} according to the rule s=(−1)^(x). The modulationformat may include an arbitrary phase offset e^(iφ).; this property ismore fully described in the paper by A. R. Hammons Jr. and H. El Gamal.On the Theory of Space-Time Codes for PSK Modulation. IEEE Trans. Info.Theory, March 2000; which is incorporated herein by reference in itsentirety. Notationally, the circled operator ⊕ is used to distinguishmodulo 2 addition from real- or complex-valued (+, −) operations.

The base-band rank criterion does not allow for a systematic approachfor designing algebraic space-time codes because it applies to thecomplex domain rather than the discrete domain in which codes aretraditionally designed. The following binary rank criterion is employedto aid the design of algebraic full diversity space-time codes for BPSKmodulation; this binary rank criterion is further detailed in a paper byA. R. Hammons Jr. and H. El Gamal, entitled “On the Theory of Space-TimeCodes for PSK Modulation”, IEEE Trans. Info. Theory, March 2000; whichis incorporated herein by reference it is entirety. With respect to thebinary rank criterion, it is assumed that C is a linear L_(t)×nspace-time code with underlying binary code C of length N=nL_(t), wheren≧L_(t). It is also assumed that every non-zero code word c is a matrixof full rank over the binary field F. Thus, for BPSK transmission overthe quasi-static fading channel, the space-time code C achieves fullspatial transmit diversity L_(t).

Next, assuming C^((s)) denotes the rate k/n binary convolutional codethat is used in the single antenna system. The encoder 203 processes kbinary input sequences x₁(t),x₂(t), . . . ,x_(k)(t) and produces n codedoutput sequences y₁ ^((s))(t),y₂ ^((s))(t), . . . ,y_(n) ^((s))(t) whichare multiplexed together to form the output code word. A sequence {x(t)}is often represented by the formal series X(D)=x(0)+x(1)D+x(2)D²+ . . .. {x(t)}<−>X(D), which is referred to as a D-transform pair. The actionof the binary convolutional encoder 203 is linear and is characterizedby the so-called impulse responses g_(i,1) ^((s))(t)<−>G_(i,j) ^((s))(D)associating output y_(j) ^((s))(t) with input x_(i)(t). Thus, theencoder action is summarized by the matrix equation:Y ^((s))(D)=X(D)G ^((s))(D),

where Y^((s))(D)=└Y₁ ^((s))(D)Y₂ ^((s))(D) . . . Y_(n) ^((s))(D)┘,X(D)=[X₁(D)X₂(D) . . . X_(k)(D)], and ${G^{(s)}(D)} = {\begin{bmatrix}{G_{1,1}^{(s)}(D)} & {G_{1,2}^{(s)}(D)} & \ldots & {G_{1,n}^{(s)}(D)} \\{G_{2,1}^{(s)}(D)} & {G_{2,2}^{(s)}(D)} & \ldots & {G_{2,n}^{(s)}(D)} \\\vdots & \vdots & ⋰ & \vdots \\{G_{k,1}^{(s)}(D)} & {G_{k,2}^{(s)}(D)} & \ldots & {G_{k,n}^{(s)}(D)}\end{bmatrix}.}$

The space-time overlay code C in which the code word Y^((i))(D) istransmitted from antenna i is obtained through the action of a rate k/nconvolutional encoder 203 with transfer function G^((i))(D) on thek-tuple information stream X(D). It is noted that the overlay constraint(1) is satisfied if and only if G⁽¹⁾(D)=G^((s))(D).

The following proposition establishes sufficient conditions on G⁽¹⁾(D),. . . ,G^((L) ^(t) ⁾(D) which guarantee that the space-time overlayachieves full spatial transmit diversity L_(t). With respect to a BPSKoverlay construction, G⁽¹⁾(D),G(²)(D), . . . ,G^((L) ^(t) ⁾(D) aretransfer functions for rate k/n convolutional codes, n≧k; C is a L_(t)×nspace-time code of dimension k that includes the following code words${{C(D)} = \begin{bmatrix}{{X(D)}\quad{G^{(1)}(D)}} \\{{X(D)}\quad{G^{(2)}(D)}} \\\vdots \\{{X(D)}\quad{G^{(L_{t})}(D)}}\end{bmatrix}},$where X(D) denotes the formal series of k arbitrary binary informationsequences and L_(t)≦n. C thus satisfies the binary rank criterion, andconsequently, for BPSK transmission over the quasi-static fadingchannel, achieves full spatial transmit diversity L_(t), if and only ifG⁽¹⁾(D),G⁽²⁾(D), . . . ,G^((L) ^(t) ⁾(D) have the property that∀α₁,α₂, . . . α_(L) _(t) ∈F:G(D)=α₁ G ⁽¹⁾(D)⊕α₂ G ⁽²⁾(D)⊕ . . . ⊕α_(L) _(t) G ^((L) ^(t) ⁾(D)is of rank k over F[[D]] (the space of all formal series) unless α₁=α₂=. . . =α_(L) _(t) =0.

Assuming G(D) has rank k over F[[D]], then, for(X)(D)G(D)=X(D)└α₁ G ⁽¹⁾(D)⊕α₂ G ⁽²⁾(D)⊕ . . . ⊕α_(L) _(t) G ^((L) ^(t)⁾(D)┘to be equal to 0, one of the following conditions, X(D)=0 or α₁=α₂= . .. α_(L) _(t) =0, needs to be satisfied. Hence, C satisfies the binaryrank criterion.

Further, it is assumed that G(D) has rank less than k over F[[D]].Consequently, there is a nonzero X(D) such thatX(D)G(D)=α₁ X(D)G ⁽¹⁾(D)⊕α₂ X(D)G ⁽²⁾(D)⊕ . . . ⊕α_(L) _(t) X(D)G ^((L)^(t) ⁾(D)=0for α₁=α₂= . . . α_(L) _(t) =0 other than the all zero case. Hence, Cdoes not satisfy the binary rank criterion.

For the special case in which C^((s)) is a rate 1/n convolutional code,it is sufficient to choose G_(j) ⁽¹⁾, . . . ,G_(j) ^((L) ^(t) ⁾ for anysingle arbitrary j, 1≦j≦n, according to the stacking constructionproposed by A. R. Hammons Jr. and H. El Gamal in “On the Theory ofSpace-Time Codes for PSK Modulation” (IEEE Trans. Info. Theory, March2000). Use of such a stacking construction ensures that the resultingspace-time code achieves full diversity. However, it is more intuitivelyappealing to construct G_(j) ⁽¹⁾, . . . ,G_(j) ^((L) ^(t) ⁾ for all j,1≦j≦n , according to this stacking construction.

Except for the constraint that G⁽¹⁾(D)=G^((s))(D), no upper bounds areimposed on the constraint lengths of the other transfer functionsG⁽²⁾(D), . . . ,G^((L) ^(t) ⁾(D). However, restricting these constraintlengths limits the trellis complexity of the overall space-time code.More specifically, Viterbi decoding can be leveraged for the singleantenna code C^((s)) by limiting the maximum constraint length ofG⁽¹⁾(D), . . . ,G^((L) ^(t) ⁾(D) to be equal to that of G^((s))(D). Inthis manner, the resulting space-time code has the same trelliscomplexity as G^((s))(D), and the only modification involves changingthe branch metric computations of the single antenna Viterbi decoder.The branch metric computations depend on the number of transmit antennas207 and receive antennas 303 [1].

In the case of QPSK modulation, the natural discrete symbol alphabet Yis the ring Z₄={0, ±1, 2} of integers modulo 4. Modulation is performedby mapping the symbol x ∈ Z₄ to the constellation point s ∈ {±1, ±i}according to the rule s=i^(x), where i=√{square root over (−1)}. It isnoted that the absolute phase reference of the QPSK constellation may bechosen arbitrarily without affecting the performance. Since the binaryrank criterion developed by A. R. Hammons Jr. and H. El Gamal in “On theTheory of Space-Time Codes for PSK Modulation” (IEEE Trans. Info.Theory, March 2000) for QPSK modulated space-time codes pertains tocertain projections of the Z₄-valued matrix c over the binary field, thefollowing definitions are stated. First, c is defined as a Z₄-valuedmatrix that includes l rows and p columns, which are not multiples oftwo; after suitable row permutations if necessary, the matrix has thefollowing row structure: $c = \begin{bmatrix}{\underset{\_}{c}}_{1} \\\vdots \\{\underset{\_}{c}}_{l} \\{2c_{l + 1}^{\prime}} \\\vdots \\{2c_{L_{t}}^{\prime}}\end{bmatrix}$The row-based indicant projection (Ξ-projection) is then defined asfollows: ${\Xi\quad c} = \begin{bmatrix}{\beta\quad\left( {\underset{\_}{c}}_{1} \right)} \\\vdots \\{\beta\quad\left( {\underset{\_}{c}}_{l} \right)} \\{\beta\quad\left( c_{l + 1}^{\prime} \right)} \\\vdots \\{\beta\quad\left( c_{L_{t}}^{\prime} \right)}\end{bmatrix}$where β(c _(i)) is the binary projection of the Z₄ vector c _(j).Similarly, the column-based indicant projection (Ψ-projection) isdefined as[Ψ(c)]^(T)=[Ξ(c)]^(T)   (3)The row and column indicant projections serve to indicate certainaspects of the binary structure of the Z₄ matrix in which multiples oftwo are ignored. Using these binary indicants, the following binary rankcriterion for QPSK modulated codes is created.

In the QPSK binary rank criterion, C denotes a linear L_(t)×n space-timecode over Z₄ with n≧L_(t). For every non-zero C ∈ C, the row-basedindicant Ξ(c) or the column-based indicant Ψ(c) has full rank L_(t) overF. Consequently, for QPSK transmission, the space-time code C achievesfull spatial diversity L_(t).

In conventional single antenna communication systems, binaryconvolutional codes with optimal free distances d_(free) are used. Theencoder output is then mapped to the Z₄ alphabet according to the Graymapping rule (i.e., 00→0, 01→1, 11→2, 10→3). The resulting codemaximizes the minimum Hamming distance between any two distinct codewords and, hence maximizes the minimum Euclidean distance among theclass of codes based on binary convolutional codes.

Taking advantage of the structure of the single dimensional binary codein designing QPSK space-time overlays introduces greater complexity thanthe BPSK scenario due to the non-linearity of the Gray mapped binarycode over the Z₄ ring of integers. In this case, the QPSK binary rankcriterion only applies to differences between code words which increasethe difficulty involved in extracting an algebraic framework forconstructing overlays. Therefore, the use of systematic inner space-timecodes that satisfy the overlay constraint and achieve full spatialdiversity are employed, according to one embodiment of the presentinvention. The stacking construction, as described in the paper by A. R.Hammons Jr. and H. El Gamal, entitled “On the Theory of Space-Time Codesfor PSK Modulation”, IEEE Trans. Info. Theory, March 2000, can be thebasis for constructing systematic inner block or convolutional codeachieving full diversity.

It is instructive to discuss the design principle of the innerconvolutional code. The coded Z₄ output stream X^(z) ⁴ (D) after Graymapping is presented at the input of the inner Z₄ rate 1/L_(t)convolutional code C^(z) ⁴ (D) with the following Z₄ transfer functionG ^((z) ⁴ ⁾(D)=└G ₁ ^((z) ⁴ ⁾(D)G ₁ ^((z) ⁴ ⁾(D) . . . G _(L) _(t) ^((z)⁴ ⁾(D)┘  (4)In the natural space-time formatting of C^(z) ⁴ (D), the output sequencecorresponding to Y_(i) ^((z) ⁴ ⁾(D)=X^((z) ⁴ ⁾(D)G_(i) ^((z) ⁴ ⁾(D) isassigned to the i-th transmit antenna 200. This construction satisfiesthe overlay constraint if and only if C^(z) ⁴ (D) is a systematic code(i.e., G₁ ^((z) ⁴ ⁾(D)=1). The resulting space-time code C satisfies theQPSK binary rank criterion under relatively mild conditions on thegenerator polynomials.

Turning now to the construction of the QPSK overlay, it is assumed thatG_(c) is the Z₄ coefficients matrix corresponding to the naturalspace-time code C associated with the rate 1/L_(t) non-recursiveconvolutional code C^(z) ⁴ (D). Accordingly, C satisfies the QPSK binaryrank criterion, and thus achieves full spatial transmit diversity L_(t)for QPSK transmission, if the binary projection β(G_(c)) has full rankL_(t) as a matrix of coefficients over the binary field F.

It is noted that one general delay diversity transmission format is aspecial case of the QPSK overlay construction. Since the condition inthe QPSK overlay construction is related to the binary projection of thetransfer function, the linear Z₄ codes can be obtained by lifting fulldiversity binary convolutional codes to the Z₄ domain (i.e., each 1 inthe binary code coefficients matrix can be replaced with either 1 or 3and each 0 with either 0 or 2). Binary rate 1/L_(t) convolutional codeswith optimal d_(free) are good candidates for this application as theirassociated natural space-time codes typically satisfy the binary rankcriteria. Furthermore, these codes have been observed to outperform thebest conventional space-time trellis codes, as determined by extensivecomputer search methods, especially for increasing numbers of antennas.This study is documented by H. El Gamal and A. R. Hammons Jr. (AlgebraicDesigns for Coherent and Differentially Coherent Space-Time Codes.Presented at the WCNC, 2000; which is incorporated herein by referencein its entirety). The desired full diversity inner systematic codes canbe obtained by lifting the recursive version of those optimal freedistance codes to the Z₄ domain.

Joint maximum likelihood decoding of the outer single dimensional codeC^((s)) and the inner systematic space-time code C introducessignificant complexity, especially for large number of transmit antennas207, due to the large number of states in the joint trellis diagram.Fortunately, this does not impose a major obstacle since this codingscheme allows for a straightforward application of the turbo processingarchitecture [3]. A soft input/soft output decoder can be used for bothC^((s)) and C, and the decoding process should be iterated with softinformation passing between the two decoder. A random interleaver may beused to scramble the output stream of C^((s)) before passing it to C.This is necessary to aid the turbo decoder convergence, and does notaffect the diversity advantage achieved by the inner space-time code.Guided by the excellent performance exhibited by this architecture invarious applications, it is expected that this receiver offersperformance that is very close to maximum likelihood decoding withreasonable complexity.

The added complexity that is required to decode the space-time overlayconstruction, as described above, can be avoided when C^((s)) is a rate1/n convolutional code. In this special case, a space-time overlayconstruction is provided with the same trellis complexity as that ofC^((s)). For the purpose of explanation, the case wherein C^((s)) is arate 1/2 binary convolutional code is considered. The extension toarbitrary rate 1/n codes is then described. It is assumed that the twooutput branches from the encoder Y₁ ^((s))(D), Y₂ ^((s))(D) are groupedaccording to the Gray mapping rule to form the Z₄ stream Y_(z) ₄^((s))(D). The only implication of this assumption is that temporalinterleaving has to be performed on a QPSK symbol by symbol basis. Basedon the Gray mapping rule, the following relation exists:Y _(z) ₄ ^((s))(D)=(Y ₁ ^((s))(D)⊕Y ₂ ^((s))(D))+2Y ₂ ^((s))(D)   (5)and hence,β(Y _(z) ₄ ^((s))(D))=Y ₁ ^((s))(D)⊕Y ₂ ^((s))(D)=X(D)(G ₁ ^((s))(D)⊕G ₂^((s))(D))   (6)Therefore the binary projection of the Z₄ stream is equivalent to a rate1/2 convolutionally encoded stream with the generator polynomial G₁⁽²⁾(D)⊕G₂ ^((s))(D). This observation leads to a second overlayconstruction, as described below.

In this second QPSK overlay construction, it is assumed that C is a Z₄L_(t)×n space-time code obtained by grouping the two output branchesfrom L_(t) rate 1/2 binary convolutional encoders G⁽¹⁾(D), . . . ,G^((L)^(t) ⁾(D) according to the Gray mapping rule. Then, for QPSKtransmission over the quasi-static fading channel, C satisfies the QPSKbinary rank criterion, and hence achieves full spatial diversity if∀α₁,α₂, . . . ,α_(L) _(t) ∈ F:α₁(G ₁ ⁽¹⁾(D)⊕G ₂ ⁽¹⁾(D))⊕α₂(G ₁ ⁽²⁾(D)⊕G ₂ ⁽²⁾(D))⊕ . . . ⊕α_(L) _(t)(G ₁ ^((L) ^(t) ⁾(D)⊕G ₂ ^((L) ^(t) ⁾(D))≠0unless α₁=α₂= . . . α_(L) _(t) =0.

It is assumed that α₁(G₁ ⁽¹⁾(D)⊕G₂ ⁽¹⁾(D))⊕α₂(G₁ ⁽²⁾(D)⊕G₂ ⁽²⁾(D))⊕ . .. ⊕α_(L) _(t) (G₁ ^((L) ^(t) )(D)⊕G₂ ^((L) ^(t) ⁾(D))≠0 unless α₁=α₂= .. . α_(L) _(t) =0. Based on the Gray mapping rule, the two outputbranches from the encoder Y₁ ^((i))(D) and Y₂ ^((i))(D) that correspondto antenna i, i=1, 2, . . . ,L_(t), are grouped to yieldY _(z) ₄ ^((i))(D)=(Y ₁ ^((i))(D)⊕Y ₂ ^((i))(D))+2Y ₂ ^((i))(D).The binary projection of Y_(z) ₄ ^((i))(D) isβ(Y _(z) ₄ ^((i))(D))=Y ₁ ^((i))(D)⊕Y ₂ ^((i))(D)=X(D)G ₁ ^((i))(D)⊕G ₂^((i))(D))for i=1, 2, . . . ,L_(t). Therefore, the row-based indicant projectionis given by ${\Xi\quad\left( {C\quad(D)} \right)} = {\begin{bmatrix}{\beta\quad\left( {Y_{z_{4}}^{(1)}(D)} \right)} \\{\beta\quad\left( {Y_{z_{4}}^{(2)}(D)} \right)} \\\vdots \\{\beta\quad\left( {Y_{z_{4}}^{(L_{t})}(D)} \right)}\end{bmatrix} = {\begin{bmatrix}{X\quad(D)\left( {{G_{1}^{(1)}(D)} \oplus {G_{2}^{(1)}(D)}} \right)} \\{X\quad(D)\left( {{G_{1}^{(2)}(D)} \oplus {G_{2}^{(2)}(D)}} \right)} \\\vdots \\{X\quad(D)\left( {{G_{1}^{(L_{t})}(D)} \oplus {G_{2}^{(L_{t})}(D)}} \right)}\end{bmatrix}.}}$Now, ∀α₁,α₂, . . . ,α_(L) _(t) ∈ F:a₁β  (Y_(z₄)⁽¹⁾(D)) ⊕ a₂β  (Y_(z₄)⁽²⁾(D)) ⊕ … ⊕ a_(L_(t))β  (Y_(z₄)^((L_(t)))(D)) = a₁X  (D)(G₁⁽¹⁾(D) ⊕ G₂⁽¹⁾(D)) ⊕ … ⊕ a_(L_(t))X  (D)(G₁^((L_(t)))(D) ⊕ G₂^((L_(t)))(D)) = X  (D)  a₁⌊(G₁⁽¹⁾(D) ⊕ G₂⁽¹⁾(D)) ⊕ … ⊕ a_(L_(t))(G₁^((L_(t)))(D) ⊕ G₂^((L_(t)))(D))⌋ ≠ 0unless X(D)=α₁=α₂= . . . α_(L) _(t) =0. Hence, Ξ(C(D)) has full rank forall non-zero code words. Therefore, C satisfies the QPSK binary rankcriterion, and hence, achieves full spatial diversity.

The above second overlay construction implies that it is sufficient tochoose G₁ ⁽¹⁾(D)⊕G₂ ⁽²⁾(D), . . . ,G₁ ^((L) ^(t) ⁾(D)⊕G₂ ^((L) ^(t) ⁾(D)to ensure that the Z₄ code achieves full diversity. By restricting themaximum constraint length of any component in G(D) to be equal to thatof C^((s)), it is readily observed that C has the same trelliscomplexity as C^((s)). This second QPSK overlay construction can beeasily extended to construct space-time overlays for systems with rate1/2m codes. For rate 1/(2m+1) codes, the condition in this overlayconstruction is slightly modified; in this case, the code needs to berepresented in a rate 2/2(2m+1) form, wherein the condition for fulldiversity is that all the linear combinations of the 2×(2m+1) transferfunctions resulting from the binary projection operator β must have fullrank 2 over the space of all formal series. The trellis diagram of thenew representation has four branches coming out of each state; however,the number of branches per decoding bit remains the same as that in thesingle dimensional code C^((s)).

The algebraic framework previously discussed encompasses a wide range ofconvolutional based space-time overlays. All space-time codes withinthis framework achieve full spatial diversity. The second criterion thatdetermines the performance of space-time codes in quasi-static fadingchannels is the product distance (coding advantage) which does notaffect the asymptotic slope, but results in a shift of the asymptoticperformance curve.

In quasi-static fading channels, the product distance η of a space-timecode C is defined as the minimum over all distinct pairs of code wordsc, e ∈ C, of the geometric mean of the eigenvalues ofA=(ƒ(c)−ƒ(e))(ƒ(c)−ƒ(e))^(H). The upper bound on the product distance ofthe class of linear BPSK space-time codes is now derived. C is a linearfull diversity L_(t)×n space-time code with underlying binary code C oflength N=nL_(t), where n≧L_(t), and free distance d_(free). Then, forBPSK transmission over the quasi-static fading channel, the space-timecode product distance η is upper bounded by 4d_(free)/L_(t) (i.e.,η≦4d_(free)/L_(t)). Assuming λ₁, . . . ,λ_(L) _(t) be the eigenvalues ofthe full rank matrix A=(ƒ(c)−ƒ(e))(ƒ(c)−ƒ(e))^(H), then $\begin{matrix}{{{\sum\limits_{j = 1}^{L_{t}}\quad\lambda_{j}} = {{trA} = {4d_{e,c}}}},} & (7)\end{matrix}$where d_(e,c) is the binary distance between the code words e, c ∈ C,and $\begin{matrix}{{\min\limits_{c,{e \in C}}{\sum\limits_{j = 1}^{L_{t}}\quad\lambda_{j}}} = {4{d_{free}.}}} & (8)\end{matrix}$Subject to this constraint on the sum of the eigenvalues, the productdistance obtained by the optimal parsing function is upper bounded by$\begin{matrix}{{\eta \leq \lambda_{1}^{o}} = {\lambda_{2}^{o} = {\ldots = {\lambda_{L_{t}}^{o} = {\frac{4d_{free}}{L_{t}}.}}}}} & (9)\end{matrix}$

Orthogonal space-time codes are particularly appealing because of thesimplicity of their maximum likelihood decoder [2]. This simplicity is aresult of the orthogonality between the rows of the space-time code wordmatrix c. It is readily observed that using a slightly modified versionof the real orthogonal space-time codes [2]—in which some of the columnsare multiplied by (−1) to adjust the sign of the first entry—as innerappliques to upgrade single antenna BPSK modulated systems satisfies theoverlay constraint in expression (1). The following result establishesthe product distance that can be achieved by this overlay design, whichrivals that of the optimal convolutional based space-time overlay withthe same constraint length. Interestingly, this near optimal performanceis also facilitated by the orthogonality between the rows of theresulting space-time code.

It is assumed that C is a full diversity L_(t)×n concatenated space-timecode with single dimensional outer code C^((s)) of length n and innerorthogonal block code of length L_(t), and d_(free) ^((s)) is the freedistance of C^((s)). Then, for BPSK transmission over the quasi-staticfading channel, the product distance of C is η=d_(free) ^((s)). Theorthogonality between the different rows of (ƒ(c)−ƒ(e))results in adiagonal A=(ƒ(c)−ƒ(e))(ƒ(c)−ƒ(e))^(H) for all distinct pairs of codewords c, e ∈ C. Hence, for the code ∈C^((s)) with the minimum distanceseparation d_(free) ^((s)), the following expression results:η=λ₁=λ₂= . . . λ_(L) _(t) =4d _(free) ^((s))   (10)

The product distance achieved by the concatenated coding approach andthe derived upper bound are compared below in Table 1, for someexemplary scenarios; in particular, a BPSK system with rate 1/2 singledimensional code and optimal free distance. TABLE 1 L_(t) K η UpperBound 2 3 20 20 2 4 24 26 2 5 28 32 2 6 32 36 2 7 40 40 3 3 20 21 3 4 2426 3 5 28 32 3 6 32 36 3 7 40 45In the comparison of Table 1, the constraint lengths of C and C^((s))are the same to allow for the same decoder complexity. In all consideredcases, it is shown that the concatenated coding approach achieves eitheroptimal or very near optimal performance. It is also worth noting thatthe same optimality argument for this overlay design approach holds forQPSK modulated systems with only two transmit antennas 207 inquasi-static fading channels.

In block fading channels, the code word is composed of multiple blocks.The fading coefficients are constant over one fading block, but areindependent from block to block. The number of facing blocks per codeword M can be regarded as a measure of the interleaving delay allowed inthe system 100, so that systems subject to a strict delay constraint areusually characterized by a small number of independent blocks [4].

The framework developed above for the quasi-static fading channel,according to the present invention, can be extended to block fadingchannels using the machinery introduced in (2). The objective in thisscenario is to exploit both temporal and spatial diversity available inthe system. In such channels, the maximum transmit diversity advantagepossible with space-time overlays (without factoring in the effect ofthe receive antennas 303) is given by the following expression:$\begin{matrix}{{d_{m} = {\left\lbrack {L_{t}{M\left( {1 - \frac{r}{L_{t}{\Omega }}} \right)}} \right\rbrack + 1}},} & (11)\end{matrix}$where L_(t) is the number of transmit antennas 207, M is the number offading blocks per code word, r is the transmission rate, and is the sizeof the constellation alphabet. It is interesting to compare this resultwith the maximum diversity advantage possible for the single antennasystem supporting the same transmission throughput [5]: $\begin{matrix}{{d_{s} = {\left\lbrack {M\left( {1 - \frac{r}{\Omega }} \right)} \right\rbrack + 1}},} & (12)\end{matrix}$where it is clear that d_(m)≧L_(t)×d_(s). This inequality suggests thatdesign approaches that are optimized for quasi-static fading channelsmay not yield the maximum possible diversity advantage for block fadingchannels. The primary example is the concatenated coding approach withinner block orthogonal space-time codes discussed previously. Thisapproach yields excellent performance in quasi-static fading channels,but suffers degradation in performance in block fading channels. Thereason for the degradation is that the simple maximum likelihood decoderdictates that the transmission of a complete inner code word be in thesame fading state [2]. This limits the maximum possible diversityadvantage to d_(cone)=L_(t)×d_(s).

Table 2 compares d_(cone), d_(m) for some exemplary scenarios,illustrating the possible diversity advantages of the algebraic overlayapproach and the concatenated coding approach in a BPSK system with 0.5bps/Hz. TABLE 2 L_(t) M d_(conc) d_(rn) 2 1 2 2 2 2 4 4 2 3 4 5 2 4 6 72 5 6 8 3 1 3 3 3 2 6 6 3 3 6 8 3 4 9 11 3 5 9 13

Next, the results for the algebraic space-time overlays forconvolutionally coded systems are discussed. The search results foralgebraic space-time overlays obtained from underlying rate 1/2 and rate1/3 convolutional codes are presented. In particular, Table 3 showsalgebraic overlays for systems with underlying single dimensional rate1/2 convolutional codes.

Table 3 considers the following parameters: L_(t)=1, 2, 3 transmitantennas 207, and convolutional codes with constraint lengths of K=4, .. . ,7. All codes achieve full diversity for both BPSK and QPSKtransmissions (with Gray mapping) over the quasi static fadingchannel—i.e. they satisfy the BPSK and QPSK rank criteria. Furthermore,with the exception of the constraint length K=5 convolutional code whichachieves free distance Of d_(free)−1, these codes achieve optimal valuesof the free distance d_(free). TABLE 3 K = ν + 1 L_(t) = 1 L_(t) = 2L_(t) = 3 4 15, 17 15, 17, 13, 15 15, 17, 13, 15, 17, 13* 5 23, 35 23,35, 25, 37 23, 35, 25, 37, 27, 33 6 53, 75 53, 75, 67, 71 53, 75, 67,71, 55, 57 7 133, 171 133, 171, 117, 165 133, 171, 117, 165, 151, 137

In Table 4, space-time overlays are obtained for systems with underlyingrate 1/3 convolutional codes, with the following parameters: L_(t)=1, 2transmit antennas 207, and convolutional codes with constraint lengthsK=4, . . . ,7. All codes provide optimal values of d_(free) whileachieving full diversity for both BPSK and QPSK transmissions. TABLE 4 K= ν + 1 L_(t) = 1 L_(t) = 2 4 13, 15, 17 13, 15, 17, 17, 13, 15 5 25,33, 37 25, 33, 37, 35, 27, 35 6 47, 53, 75 47, 53, 75, 65, 57, 73 7 133,145, 175 133, 145, 175, 175, 175, 133 8 225, 331, 367 225, 331, 367,277, 263, 355

FIGS. 4A-4F show the simulation results for the algebraic convolutionalspace-time overlays, in accordance with an embodiment of the presentinvention. These results demonstrate the excellent performance achievedby the codes of the present invention and quantify the possibleimprovements with increasing numbers of transmit antennas 207. In allthe examples, one frame corresponds to 130 transmissions for allantennas 207. The scenario with rate 1/2 single dimensional code isconsidered, in which the system 100 achieves a spectral efficiency of0.5 and 1 bits/sec/Hz in the case of BPSK and QPSK modulation,respectively.

In FIGS. 4A-4C, a BPSK modulated system is considered. FIG. 4A providesperformance comparisons for the constraint length 7 algebraic space-timeoverlays with one, two, and three transmit antennas 207. The number ofreceive antennas 303 is one in the three cases. It is observed that at aframe error rate (FER) of 10⁻¹ the systems with two and three transmitantennas 207 provide gains of approximately 3 and 5 dB over theunderlying single antenna system. At a FER of 10⁻², the gains of theconvolutional space-time overlays with two and three antennas comparedto the single antenna system are even higher: 8 dB and 10.5 dB,respectively. In FIG. 4B, the performances of space-time overlays withdifferent constraint lengths are compared for a system with two transmitantennas 207 and two receive antennas 303. The performance ofconvolutional space-time codes is shown to improve as the constraintlength of the code increases. For example, the constraint length K=9convolutional code outperforms the constraint length K=5 code by 1.5 dB.FIG. 4C compares the performance of the algebraic convolutionalspace-time overlay and that of the concatenated coding approach withinner orthogonal codes. In this particular scenario, it is observed thatboth approaches provide identical performance.

The same comparisons are then repeated for QPSK modulated systems, perFIGS. 4D-4F. In FIG. 4D, the gain obtained by increasing the number oftransmit antennas 207 when algebraic space-time overlays are used isquantified. At a frame error rate (FER) of 10⁻¹, the systems with twoand three transmit antennas 207 provide gains of approximately 3 and 4.5dB, whereas at a FER of 10⁻², the gains increase to 7.5 dB and 10 dB,respectively. FIG. 4E compares the performance of space-time overlayswith different constraint lengths in a system with two transmit antennas207 and two receive antennas 303. It is shown that the constraint lengthK=9 space-time code outperforms the constraint length K=5 code by 1.5dB—similar to the BPSK scenario. Finally, the performance of the K=5 andK=7 algebraic overlays is compared in FIG. 4F for the case of threetransmit antennas 207 and three receive antennas 303, where it is shownthat the K=7 convolutional code outperforms the K=5 code by 1 dB at aFER of 10⁻².

The above codes, according to the present invention, have applicabilityin a number of communication systems; for example, the space-time codescan be deployed in a wireless communication, as seen in FIG. 5.

FIG. 5 shows a diagram of a wireless communication system that utilizesspace-times according to an embodiment of the present invention. In awireless communication system 500, multiple terminals 501 and 503communicate over a wireless network 505. Terminal 501 is equipped with aspace-time encoder 203 (as shown in FIG. 2) that generates the overlayspace-time codes. Terminal 501 also includes multiple transmit antennas207 (as shown in FIG. 2). In this example, each of the terminals 501 and503 are configured to encode and decode the space-time codes;accordingly, both of the terminals 501 and 503 possess the transmitter200 and receiver 300. However, it is recognized that each of theterminals 501 and 503 may alternatively be configured as a transmittingunit or a receiving unit, depending on the application. For example, ina broadcast application, terminal 501 may be used as a head-end totransmit signals to multiple receiving terminals (in which onlyreceiving terminal 503 is shown). Consequently, terminal 503 would onlybe equipped with a receiver 300. The space-time code construction of thepresent invention advantageously permits use of a smaller number ofreceive antennas 303 than that of the transmitting terminal 501, therebyresulting in hardware cost reduction. In an exemplary embodiment, aterminal that is designated as a receiving unit may possess a smallerquantity of antennas that of the transmitting unit.

FIG. 6 shows a diagram of a computer system that can perform theprocesses of encoding and decoding of space-time codes, in accordancewith an embodiment of the present invention. Computer system 601includes a bus 603 or other communication mechanism for communicatinginformation, and a processor 605 coupled with bus 603 for processing theinformation. Computer system 601 also includes a main memory 607, suchas a random access memory (RAM) or other dynamic storage device, coupledto bus 603 for storing information and instructions to be executed byprocessor 605. In addition, main memory 607 may be used for storingtemporary variables or other intermediate information during executionof instructions to be executed by processor 605. Computer system 601further includes a read only memory (ROM) 609 or other static storagedevice coupled to bus 603 for storing static information andinstructions for processor 605. A storage device 611, such as a magneticdisk or optical disk, is provided and coupled to bus 603 for storinginformation and instructions.

Computer system 601 may be coupled via bus 603 to a display 613, such asa cathode ray tube (CRT), for displaying information to a computer user.An input device 615, including alphanumeric and other keys, is coupledto bus 603 for communicating information and command selections toprocessor 605. Another type of user input device is cursor control 617,such as a mouse, a trackball, or cursor direction keys for communicatingdirection information and command selections to processor 605 and forcontrolling cursor movement on display 613.

According to one embodiment, interaction within system 100 is providedby computer system 601 in response to processor 605 executing one ormore sequences of one or more instructions contained in main memory 607.Such instructions may be read into main memory 607 from anothercomputer-readable medium, such as storage device 611. Execution of thesequences of instructions contained in main memory 607 causes processor605 to perform the process steps described herein. One or moreprocessors in a multi-processing arrangement may also be employed toexecute the sequences of instructions contained in main memory 607. Inalternative embodiments, hard-wired circuitry may be used in place of orin combination with software instructions. Thus, embodiments are notlimited to any specific combination of hardware circuitry and software.

Further, the instructions to support the generation of space-time codesof system 100 may reside on a computer-readable medium. The term“computer-readable medium” as used herein refers to any medium thatparticipates in providing instructions to processor 605 for execution.Such a medium may take many forms, including but not limited to,non-volatile media, volatile media, and transmission media. Non-volatilemedia includes, for example, optical or magnetic disks, such as storagedevice 611. Volatile media includes dynamic memory, such as main memory607. Transmission media includes coaxial cables, copper wire and fiberoptics, including the wires that comprise bus 603. Transmission mediacan also take the form of acoustic or light waves, such as thosegenerated during radio wave and infrared data communication.

Common forms of computer-readable media include, for example, a floppydisk, a flexible disk, hard disk, magnetic tape, or any other magneticmedium, a CD-ROM, any other optical medium, punch cards, paper tape, anyother physical medium with patterns of holes, a RAM, a PROM, and EPROM,a FLASH-EPROM, any other memory chip or cartridge, a carrier wave asdescribed hereinafter, or any other medium from which a computer canread.

Various forms of computer readable media may be involved in carrying oneor more sequences of one or more instructions to processor 605 forexecution. For example, the instructions may initially be carried on amagnetic disk of a remote computer. The remote computer can load theinstructions relating to encoding and decoding of space-time codes usedin system 100 remotely into its dynamic memory and send the instructionsover a telephone line using a modem. A modem local to computer system601 can receive the data on the telephone line and use an infraredtransmitter to convert the data to an infrared signal. An infrareddetector coupled to bus 603 can receive the data carried in the infraredsignal and place the data on bus 603. Bus 603 carries the data to mainmemory 607, from which processor 605 retrieves and executes theinstructions. The instructions received by main memory 607 mayoptionally be stored on storage device 611 either before or afterexecution by processor 605.

Computer system 601 also includes a communication interface 619 coupledto bus 603. Communication interface 619 provides a two-way datacommunication coupling to a network link 621 that is connected to alocal network 623. For example, communication interface 619 may be anetwork interface card to attach to any packet switched local areanetwork (LAN). As another example, communication interface 619 may be anasymmetrical digital subscriber line (ADSL) card, an integrated servicesdigital network (ISDN) card or a modem to provide a data communicationconnection to a corresponding type of telephone line. Wireless links mayalso be implemented. In any such implementation, communication interface619 sends and receives electrical, electromagnetic or optical signalsthat carry digital data streams representing various types ofinformation.

Network link 621 typically provides data communication through one ormore networks to other data devices. For example, network link 621 mayprovide a connection through local network 623 to a host computer 625 orto data equipment operated by a service provider, which provides datacommunication services through a communication network 627 (e.g., theInternet). LAN 623 and network 627 both use electrical, electromagneticor optical signals that carry digital data streams. The signals throughthe various networks and the signals on network link 621 and throughcommunication interface 619, which carry the digital data to and fromcomputer system 601, are exemplary forms of carrier waves transportingthe information. Computer system 601 can transmit notifications andreceive data, including program code, through the network(s), networklink 621 and communication interface 619.

The techniques described herein provide an approach for designingspace-time overlays. The algebraic framework to construct theseconvolutional space-time overlays that achieve full spatial diversity inquasi-static fading channels without altering the signal transmittedfrom the first antenna is developed. For BPSK modulated systems, ageneral approach for constructing space-time overlay codes with the sametrellis complexity as the code used in the single antenna system isprovided. The general approach for QPSK modulated systems involves theuse of systematic inner space-time codes that utilize separate softinput/soft output decoders at the receiver. For QPSK modulated systemsusing rate 1/n binary convolutional codes with Gray mapping, analternative space-time construction with the same trellis complexity asthe single dimensional convolutional code is developed. The space-timeoverlay codes provide improved system throughput, while minimizingreceiver complexity.

Obviously, numerous modifications and variations of the presentinvention are possible in light of the above teachings. It is thereforeto be understood that within the scope of the appended claims, theinvention may be practiced otherwise than as specifically describedherein.

REFERENCES

[1] V. Tarokh, N. Seshadri, and A. R. Calderbank. Space-Time Codes forHigh Data Rate Wireless Communication: Performance Criterion and CodeConstruction. IEEE Trans. Info. Theory, IT-44:774-765, March 1998.

[2] V. Tarokh, H. Jafarkhani, and A. R. Calderbank. Space-Time BlockCodes from Orthogonal Designs. IEEE Trans. Info. Theory,IT-45:1456-1467, July 1999.

[3] J. Hagenauer. The Turbo Principle: Tutorial Introduction and Stateof the Art. International Symposium on Turbo Codes and Related Topics,Brest-France:1-9, September 1997.

[4] E. Biglieri, G. Caire, and G. Taricco. Limiting Performance forBlock-Fading Channels with Multiple Antennas. submitted to IEEE Trans.Info. Theory, September 1999.

[5] A. Lapidoth. The Performance of Convolutional Codes on the BlockErasure Channel Using Various Finite Interleaving Techniques. IEEETrans. Info. Theory, IT-43:1459-1473, September 1997.

1. A method for transmitting encoded signals over a communicationchannel of a communication system having a plurality of transmitantennas and a plurality of receive antennas, the method comprising:receiving a message signal; and generating a code word in response tothe message signal, the code word having a construction that is basedupon a single dimensional binary code and that specifies a space-timeoverlay having a predetermined constraint. 2-85. (canceled)